1 Introduction
We study ground states of the nonlinear Schrödinger equation with potentials:
(1.1)
(
-
Δ
)
s
u
+
V
(
x
)
u
=
f
(
x
,
u
)
,
s
∈
(
0
,
1
)
,
on the whole space ℝN (see [8]). We refer to the book [9] and the paper [14] for the definitions of fractional Laplacian on ℝN and weak solutions related. The natural condition about the function V is that it is a non-negative bounded smooth function on ℝN. The assumption about the nonlinearity term f is subcritical growth on u. The exact assumptions about V and f will be given below. Equation (1.1) arises in the study of the fractional Schrödinger equation
i
∂
Ψ
∂
t
+
(
-
Δ
)
s
Ψ
=
F
(
x
,
Ψ
)
in
ℝ
N
×
ℝ
,
when the wave function Ψ is a standing wave, that is, Ψ(x,t)=e-ictu(x), where c is a constant. This equation was introduced by Laskin [18, 19] and comes from an extension of the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. Indeed, fractional spaces and nonlocal equations play a fundamental role in the investigation of several sciences such as crystal dislocation, obstacle problem, optimization, finance, phase transition, soft thin films, multiple scattering, quasi-geostrophic flows, water waves, anomalous diffusion, conformal geometry and minimal surfaces and so on.
In recent years, problems involving fractional operators are receiving a lot of attention. In [16], Frank, Jin, and Xiong have extended the fractional Sobolev inequality results of Lions [23] to cases of the half-space and bounded domains. In particular, they have showed that for the half-space and a large class of bounded domains, a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains. In [30], Servadei and Valdinoci have extended the results of some classical nonlinear analysis theorems to the case of fractional operators on the bounded domains in RN. Dávila, del Pino, Dipierro, Valdinoci [11] have studied the concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. In [14], Felmer, Quaas, and Tan have studied the existence of positive solutions with decay at infinity for the nonlinear Schrödinger equation with the fractional Laplacian of the type (1.1), and furthermore they have analyzed the regularity, decay and symmetry properties of these solutions. In [29], Secchi has studied the existence of the radially symmetric solutions of (1.1) where the nonlinearity is autonomous and does not satisfy the usual Ambrosetti–Rabinowitz condition. Ambrosio and Figueiredo [2] have investigated the existence of nontrivial ground state solutions for the fractional scalar filed equation where the nonlinearity f∈C1,β(ℝ,ℝ) has critical growth. De Souza and Araújo [12] have studied (1.1), where the potential V is unbounded and changes sign and the nonlinearity term f may be unbounded in x; moreover, they have also analyzed the multiplicity of solutions. We refer to Li and Ma [20], Ma [27, 25, 26], Cabré and Tan [7], Felmer and Torres [15] and Secchi [28] for some related results.
By definition, the ground state (1.1) is the minimal energy weak solution to (1.1). The ground state solutions correspond to the least positive critical value of the variational (energy) functional:
Φ
(
u
)
=
1
2
∫
ℝ
N
(
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
)
𝑑
x
-
∫
ℝ
N
F
(
x
,
u
)
𝑑
x
,
u
∈
X
,
where F(x,u)=∫0uf(x,t)𝑑t, and the working space X is
X
=
{
u
∈
L
loc
1
(
ℝ
N
)
:
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
<
∞
}
with norm
∥
u
∥
=
(
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
)
1
2
and inner product
〈
u
,
v
〉
:=
∫
ℝ
N
(
(
-
Δ
)
s
/
2
u
⋅
(
-
Δ
)
s
/
2
v
+
V
(
x
)
u
v
)
𝑑
x
.
For s∈(0,1), we let 2s*=2NN-2s. The important fact for this exponent is that we have the continuous Sobolev embedding
H
s
:=
H
s
(
ℝ
N
)
↪
L
2
s
*
(
ℝ
N
)
and Sobolev’s inequality: there is a uniform constant Cs such that for any u∈H˙s(ℝN) there holds
∥
u
∥
L
2
s
*
(
ℝ
N
)
≤
C
s
∥
(
-
Δ
)
s
/
2
u
∥
L
2
(
ℝ
N
)
.
Again, we refer to [9, 14] for the definitions of the spaces Hs and H˙s and compactness results related to them. To establish the existence of ground states, usually besides the growth condition on the nonlinearity and a Nehari-type condition, the following superlinear condition due to Ambrosetti and Rabinowitz [1, 33] is assumed:
There is μ>2 such that for u≠0 and x∈ℝN one has 0<μF(x,u)≤f(x,u)u.
This super-quadratic condition implies that, for some C>0, one has F(x,u)≥C|u|μ for all u∈ℝ. Instead of (AR), we may assume the following super-quadratic condition as in [24]:
lim
|
u
|
→
∞
F
(
x
,
u
)
u
2
=
∞
.
This assumption comes from the paper [21]. In this paper, we show that under a weaker and more natural version about V and f, there exists a ground state solution to (1.1). We assume two types of conditions about V. One is for the case when V is a non-negative period function. The other is about uniformly bounded functions V on ℝN. More precisely, in the period case, we need the following hypotheses.
V
(
x
)
∈
C
(
ℝ
N
,
ℝ
)
, V(x)≥0, and supℝNV(x)=1. Moreover, V(x) is 1-periodic in each of x1,x2,…,xN.
f
(
x
,
t
)
∈
C
1
is 1-periodic in each of x1,x2,…,xN. The derivative ft is a Carathéodory function and there exists C>0 such that
|
f
t
(
x
,
t
)
|
≤
C
(
1
+
|
t
|
2
s
*
-
2
)
,
lim
|
t
|
→
∞
|
f
(
x
,
t
)
|
|
t
|
2
s
*
-
1
=
0
uniformly in x∈ℝN.
f
(
x
,
t
)
=
o
(
|
t
|
)
, as |t|→0, uniformly in x.
lim
|
t
|
→
∞
F
(
x
,
t
)
t
2
=
∞
uniformly in x.
f
(
x
,
t
)
|
t
|
is strictly increasing in t∈(-∞,0)∪(0,∞).
Note that conditions (f3) and (f4) imply that f(x,u)u-2F(x,u)>0 for any u≠0 (see [31]). Below, we extend the result of Li, Wang, and Zeng [21] to the fractional case. Our first main result concerns the existence of ground state solutions to (1.1) in the case of periodic non-negative nontrivial potentials.
Theorem 1.1.
Under assumptions (V1) and (f1)–(f4), equation (1.1) has a weak solution u∈X such that Φ(u)=c*>0 and c* is defined by c*=infNΦ(u), where
𝒩
=
{
u
∈
X
∖
{
0
}
:
ψ
(
u
)
:=
〈
Φ
′
(
u
)
,
u
〉
=
0
}
.
In our case, the nontrivial function V may have some zero points, which have not been considered in previous papers. We get Theorem 1.1 by the Nehari method [10, 32, 27]. We have the existence of a ground state to (1.1) under another type of assumptions about V (see Theorem 1.2), which will be proved in Section 3.
As for a comparison to our results, we now mention a few earlier closely related results for s=1 about the existence of entire solutions of Schrödinger-type equations. When s=1, we have the classical nonlinear Schrödinger equation with potentials:
(1.2)
-
Δ
u
+
V
(
x
)
u
=
f
(
x
,
u
)
in
ℝ
N
,
u
∈
H
1
(
ℝ
N
)
,
which has been extensively studied by many authors. In [21], Li, Wang, and Zeng had studied (1.2), the existence of ground state solutions, under a more natural super-quadratic condition on f. When s=1 and V(x)≥0, there had been several papers on the studies of nonlinear Schrödinger equations such as [5, 6, 4]. Szulkin and Weth [32] had extended the previous result by assuming weaker conditions on f. In summary, their results may be stated as below. Assume the following conditions:
V
∈
C
(
ℝ
N
,
ℝ
)
and V(x)>0. Moreover, V is 1-periodic in each of x1,x2,…,xN.
f
∈
C
(
ℝ
N
×
ℝ
,
ℝ
)
, f is 1-periodic in each of x1,x2,…,xN and satisfies (f2)–(f4) and the growth restriction
|
f
(
x
,
u
)
|
≤
a
(
1
+
|
u
|
q
-
1
)
for some a>0 and 2<q<2*:=2nn-2.
Then (1.2) has a ground state solution. We remark that f need not be of class C1 and fu need not be satisfying a growth condition. We refer to Costa [10] and Szulkin and Weth [32] for more related results.
For the autonomous nonlinearity f with f(x,u)=f(u), the ground state to (1.1) is a non-negative weak solution of the following equation:
(1.3)
(
-
Δ
)
s
u
+
V
(
x
)
u
=
f
(
u
)
in
ℝ
N
.
We still assume that f(t)∈C1(R) satisfies (f1)–(f4). We assume that V∈C0(RN) satisfies condition (V2):
0
≤
inf
ℝ
N
V
(
x
)
≤
lim
|
x
|
→
∞
V
(
x
)
=
sup
ℝ
N
V
(
x
)
<
∞
,
with the zero set K:={x∈ℝN:V(x)=0} being a compact subset of ℝN.
Here, our assumptions about V is slightly weaker than that of [17]. We have the following result.
Theorem 1.2.
Under assumptions (V2) and (f1)–(f4), equation (1.3) has a weak solution u∈X, which is a ground state in the sense that Φ(u)=c*>0 and c* is defined by c*=infNΦ(u), where
𝒩
=
{
u
∈
X
∖
{
0
}
:
ψ
(
u
)
:=
〈
Φ
′
(
u
)
,
u
〉
=
0
}
.
We shall use the notation A≲B to mean that there is a uniform constant C>0 such that A≤CB. The uniform constants C or c may vary in different lines. Moreover, Lp(Ω) is the usual Lebesgue Lp space over the domain Ω⊂RN.
The rest of this paper is organized as follows. In Section 2, we prove Theorem 1.1, that is, we establish the existence of ground states of (1.1) for the periodic case (both V and f are π-periodic in x). We mention that the variational problem related to (1.1) lacks some desirable compactness properties and we use a variant of the concentration-compactness method [22, 23, 33] in Section 2. In Section 3, we consider the potential well case and we prove Theorem 1.2.
2 The Existence of Ground States in Periodic Case
We now consider Hs weak solutions of
(2.1)
(
-
Δ
)
s
u
+
V
(
x
)
u
=
f
(
x
,
u
)
on the whole space ℝN.
Recall that we are working in the Hilbert space
X
=
{
u
∈
L
loc
1
(
ℝ
N
)
:
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
<
∞
}
,
with norm
∥
u
∥
=
(
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
)
1
2
and inner product
〈
u
,
v
〉
:=
∫
ℝ
N
(
(
-
Δ
)
s
/
2
u
⋅
(
-
Δ
)
s
/
2
v
+
V
(
x
)
u
v
)
𝑑
x
.
We point out that ∥⋅∥ is equivalent to the usual Hs norm, which is the result below.
Lemma 2.1.
Assume (V1). Then ∥⋅∥ is equivalent to ∥⋅∥Hs.
Proof.
Obviously, ∥u∥≤∥u∥Hs. Let ℝN=⋃n=1∞en, where en are cubes of unit length. By the periodicity condition of V, we may fix any en. We claim that there exists a positive constant c>0 such that
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
≥
c
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
u
2
d
x
.
Otherwise, there are (uj)⊂Hs(en) such that
∫
e
n
|
(
-
Δ
)
s
/
2
u
j
|
2
+
u
j
2
d
x
=
1
and
∫
e
n
|
(
-
Δ
)
s
/
2
u
j
|
2
+
V
(
x
)
u
j
2
d
x
→
0
.
Thus, we obtain
(2.2)
∫
e
n
|
(
-
Δ
)
s
/
2
u
j
|
2
𝑑
x
→
0
,
∫
e
n
V
(
x
)
u
j
2
𝑑
x
→
0
,
∫
e
n
u
j
2
𝑑
x
→
1
.
By Poincáre’s inequality, we obtain
∫
e
n
|
u
j
-
u
¯
j
|
2
𝑑
x
≤
∫
e
n
|
(
-
Δ
)
s
/
2
u
j
|
2
𝑑
x
,
where
u
¯
j
=
1
|
e
n
|
∫
e
n
u
j
𝑑
x
.
Thus by (2.2), we have
∫
e
n
|
u
j
-
u
¯
j
|
2
𝑑
x
→
0
,
∫
e
n
|
u
¯
j
|
2
𝑑
x
→
1
,
|
u
¯
j
|
2
→
1
|
e
n
|
,
and
(2.3)
∫
e
n
V
(
x
)
u
¯
j
2
𝑑
x
→
0
.
However,
∫
e
n
V
(
x
)
u
¯
j
2
𝑑
x
=
|
u
¯
j
|
2
∫
e
n
V
(
x
)
𝑑
x
→
1
|
e
n
|
∫
e
n
V
(
x
)
𝑑
x
>
0
.
This is a contradiction to (2.3). Thus there exists a uniformly constant c>0, which is independent to the domain u and the dimension n, such that
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
≥
c
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
u
2
d
x
.
Therefore,
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
=
∑
n
=
1
∞
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
≥
c
∑
n
=
1
∞
∫
e
n
|
(
-
Δ
)
s
/
2
u
|
2
+
u
2
d
x
≥
c
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
u
2
d
x
,
that is, ∥u∥≥c∥u∥Hs. ∎
We shall use following result, which is [3, Theorem 4.9].
Lemma 2.2.
Let Ω be an open subset of RN, let fn,f∈Lp(Ω) and let fn→f in Lp(Ω), 1≤p<∞. Then there exist a subsequence (fnj) of (fn) and g∈Lp(Ω) such that fnj→f a.e. on Ω and |f|,|fnj|≤g a.e. on Ω.
Next, we need a version of Lions’s lemma. The proof of this result can be found in [14, 2, 25].
Lemma 2.3.
Assume that (un) is a bounded sequence in Hs(RN) such that for any q∈[2,2s*) and some R>0,
lim
n
→
∞
sup
y
∈
ℝ
N
∫
B
R
(
y
)
|
u
n
|
q
𝑑
x
=
0
.
Then un→0 in Lp(RN) for p∈(2,2s*).
To prove Theorem 1.1, we also need the following lemmata.
Lemma 2.4.
Assume (f1)–(f4). Then the following assertions hold:
𝒩
≠
∅
is a
C
1
-submanifold of
X
of codimension 1 , and 0∉𝒩¯.
𝒩
is a natural constraint of Φ in the sense that 0≠u∈X is a critical point of Φ if and only if u∈𝒩 and u is a critical point of Φ|𝒩.
Proof.
(i) In view of (f1) and (f2), for any ϵ>0, there exists Cϵ>0 such that
|
f
(
x
,
u
)
|
≤
ϵ
(
|
u
|
+
|
u
|
2
s
*
-
1
)
+
C
ϵ
|
u
|
p
-
1
.
Then
|
f
(
x
,
u
)
u
|
≤
ϵ
(
|
u
|
2
+
|
u
|
2
s
*
)
+
C
ϵ
|
u
|
p
for
p
∈
(
2
,
2
s
*
)
.
The functional ψ is clearly of class C1 on X. Let 0≠v∈X and we consider the function 0<t↦ψ(tv). Again, in view of (f1), (f2), and Sobolev’s embedding, we claim that ψ(tv)>0 for t>0 small. Indeed, we have
∫
f
(
x
,
t
v
)
t
v
𝑑
x
≤
|
∫
f
(
x
,
t
v
)
t
v
𝑑
x
|
≤
ϵ
(
t
2
∫
|
v
|
2
𝑑
x
+
t
2
s
*
∫
|
v
|
2
s
*
𝑑
x
)
+
C
ϵ
t
p
∫
|
v
|
p
𝑑
x
≲
ϵ
(
t
2
∥
v
∥
H
s
2
+
t
2
s
*
∥
v
∥
H
s
2
s
*
)
+
C
ϵ
t
p
∥
v
∥
H
s
p
≲
ϵ
(
t
2
∥
v
∥
2
+
t
2
s
*
∥
v
∥
2
s
*
)
+
C
ϵ
t
p
∥
v
∥
p
.
By picking 0<ϵ<1, we obtain
ψ
(
t
v
)
=
t
2
∥
v
∥
2
-
∫
f
(
x
,
t
v
)
t
v
𝑑
x
≳
t
2
(
1
-
ϵ
)
∥
v
∥
2
-
ϵ
t
2
s
*
∥
v
∥
2
s
*
-
C
ϵ
t
p
∥
v
∥
p
>
0
for all t>0 sufficiently small.
Similarly, in view of (f3), we can see the superlinearity of f(x,v) and, by (f2) and (f4), we have the fact that f(x,v)v≥0. Thus, limt→+∞ψ(tv)=-∞. Therefore, there exists t¯ such that t¯v∈𝒩. In particular, 𝒩≠∅.
Next, condition (f4) says that
(2.4)
f
′
(
x
,
t
)
>
f
(
x
,
t
)
t
for all
t
≠
0
.
We now claim that ψ′(u)≠0 for all u∈𝒩. Indeed, if u∈𝒩 is such that
ψ
′
(
u
)
⋅
ω
=
2
〈
u
,
ω
〉
-
∫
f
(
x
,
u
)
ω
𝑑
x
-
∫
f
′
(
x
,
u
)
u
ω
𝑑
x
=
0
for all ω∈X, then, by picking ω=u, it follows that
2
∥
u
∥
2
-
∫
f
(
x
,
u
)
u
𝑑
x
-
∫
f
′
(
x
,
u
)
u
2
𝑑
x
=
0
and
∫
f
(
x
,
u
)
u
𝑑
x
-
∫
f
′
(
x
,
u
)
u
2
𝑑
x
=
0
.
However, in view of (2.4),
∫
f
(
x
,
u
)
u
𝑑
x
-
∫
f
′
(
x
,
u
)
u
2
𝑑
x
=
∫
(
f
(
x
,
u
)
u
-
f
′
(
x
,
u
)
)
u
2
𝑑
x
<
0
.
This is a contradiction. Therefore, 𝒩 is a C1-submanifold of X of codimension 1.
We now show that 0∉𝒩¯. As before, (f1), (f2), and Sobolev’s embedding imply that for u∈𝒩 and ϵ>0,
∥
u
∥
2
=
∫
f
(
x
,
u
)
u
𝑑
x
≤
ϵ
(
∫
|
v
|
2
𝑑
x
+
∫
|
v
|
2
s
*
𝑑
x
)
+
C
ϵ
∫
|
v
|
p
𝑑
x
≲
ϵ
(
∥
v
∥
H
s
2
+
∥
v
∥
H
s
2
s
*
)
+
C
ϵ
∥
v
∥
H
s
p
≲
ϵ
(
∥
v
∥
2
+
∥
v
∥
2
s
*
)
+
C
ϵ
∥
v
∥
p
.
By taking ϵ=12 and recalling that u≠0 by the definition of 𝒩, we obtain
∥
u
∥
2
s
*
-
2
≳
1
2
C
ϵ
+
1
>
0
for all
u
∈
𝒩
.
Then 0∉𝒩¯.
(ii) Let 0≠u∈X be a critical point of Φ. Then
Φ
′
(
u
)
⋅
ω
=
〈
u
,
ω
〉
-
∫
f
(
x
,
u
)
ω
𝑑
x
=
0
for all
ω
∈
X
.
Choosing ω=u yields
∥
u
∥
2
-
∫
f
(
x
,
u
)
u
𝑑
x
=
0
,
that is, u∈𝒩. Conversely, if u∈𝒩 is a critical point of Φ|𝒩, then
Φ
′
(
u
)
=
λ
ψ
′
(
u
)
for some Lagrange multiplier λ∈ℝ, that is,
〈
u
,
ω
〉
-
∫
f
(
x
,
u
)
ω
𝑑
x
=
λ
[
2
〈
u
,
ω
〉
-
∫
f
(
x
,
u
)
ω
𝑑
x
-
∫
f
′
(
x
,
u
)
u
ω
𝑑
x
]
for all ω∈X. Once again, choosing ω=u and recalling that u∈𝒩, we obtain
0
=
λ
[
∫
f
(
x
,
u
)
u
𝑑
x
-
∫
f
′
(
x
,
u
)
u
2
𝑑
x
]
=
λ
∫
(
f
(
x
,
u
)
u
-
f
′
(
x
,
u
)
)
u
2
𝑑
x
.
In view of (2.4), we necessarily have λ=0 in order to not contradict the fact that u∈𝒩. Thus, u is a critical point of Φ. ∎
The following result tells us that c* is the ground state level.
Lemma 2.5.
Under the assumptions in Lemma 2.4, c*>0 and c* is the ground state level for (2.1), that is,
c
*
=
inf
{
Φ
(
u
)
:
u
≠
0
is a solution of Problem (2.1)
}
.
Proof.
We start by observing that the functional Φ has mountain-pass geometry, that is:
There exist a,δ>0 such that Φ(u)≥a if ∥u∥=δ.
There exists v0∈X such that ∥v0∥>δ and Φ(v0)≤0.
Indeed, as in Lemma 2.4, by (f1) and (f2),
|
F
(
x
,
u
)
|
≤
ϵ
(
|
u
|
2
+
|
u
|
2
s
*
)
+
C
ϵ
|
u
|
p
for
p
∈
(
2
,
2
s
*
)
,
and Sobolev’s inequality yields (with 0<ϵ<12)
Φ
(
u
)
=
1
2
∥
u
∥
2
-
∫
F
(
x
,
u
)
𝑑
x
≥
1
2
∥
u
∥
2
-
ϵ
(
∫
|
u
|
2
𝑑
x
+
∫
|
u
|
2
s
*
𝑑
x
)
-
C
ϵ
∫
|
u
|
p
𝑑
x
≳
1
2
∥
u
∥
2
-
ϵ
(
∥
u
∥
H
s
2
+
∥
u
∥
H
s
2
s
*
)
-
C
ϵ
∥
u
∥
H
s
p
≳
(
1
2
-
ϵ
)
∥
u
∥
2
-
(
ϵ
+
C
ϵ
)
∥
u
∥
2
s
*
.
Therefore, (i) holds true with ∥u∥=δ>0 sufficiently small so that the right-hand side is
a
=
(
1
2
-
ϵ
)
δ
2
-
(
ϵ
+
C
ϵ
)
δ
2
s
*
>
0
.
Similarly, as in Lemma 2.4, we obtain (ii) from the fact that, for each 0≠u∈X, we have
lim
t
→
∞
Φ
(
t
u
)
=
-
∞
.
Thus, there exists t¯ sufficiently large and v0=t¯u such that ∥v0∥>δ and Φ(v0)≤0.
We now consider the real-valued function
0
<
t
↦
Φ
(
t
u
)
=
1
2
t
2
∥
u
∥
2
-
∫
F
(
x
,
t
u
)
𝑑
x
.
In view of (f2), (f4), and (2.4), there exists a unique critical point t^=t^(u)>0 (its global maximum) such that t^(u)u∈𝒩 and maxt>0Φ(tu)=Φ(t^(u)u). Thus,
c
*
=
inf
u
∈
𝒩
Φ
(
u
)
=
inf
u
∈
X
∖
{
0
}
max
t
>
0
Φ
(
t
u
)
.
Next, by letting
Γ
:=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
0
,
Φ
(
γ
(
1
)
)
<
0
}
,
it follows from (i) and (ii) that the mountain-pass level of Φ, that is,
c
MP
:=
inf
γ
∈
Γ
sup
0
≤
t
≤
1
Φ
(
γ
(
t
)
)
is positive. Since Φ(tu)<0 for all u∈X∖{0} and t sufficiently large, we obtain
inf
γ
∈
Γ
sup
0
≤
t
≤
1
Φ
(
γ
(
t
)
)
≤
inf
u
∈
X
∖
{
0
}
max
t
>
0
Φ
(
t
u
)
.
As in [32], the manifold 𝒩 separates X into two path-connected components. By (f1) and (f2), the component containing the origin also contains a small ball around the origin. Note that Φ(u)≥0 for all u in this component. The reason is that 〈Φ′(tu),u〉≥0 for all 0≤t≤t^(u). Clearly, every path γ∈Γ has to cross 𝒩, and we have
inf
u
∈
𝒩
Φ
(
u
)
≤
inf
γ
∈
Γ
sup
0
≤
t
≤
1
Φ
(
γ
(
t
)
)
.
Thus,
c
*
=
c
MP
>
0
.
∎
We now want to show that the minimizing sequence at the level c* is bounded in Hs.
Lemma 2.6.
Assume (f1)–(f3). Then any minimizing sequence (un) for c* is bounded.
Proof.
Let (un)⊂𝒩 be a minimizing sequence for c* such that
ψ
(
u
n
)
=
∥
u
n
∥
2
-
∫
f
(
x
,
u
n
)
u
n
𝑑
x
=
0
,
(2.5)
Φ
(
u
n
)
=
1
2
∥
u
n
∥
2
-
∫
F
(
x
,
u
n
)
𝑑
x
→
c
*
.
Claim: There exists C>0 such that ∥un∥≤C for all n∈N. We argue by contradiction that (for some subsequence, still labeled un for simplicity) we have tn:=∥un∥→∞. By picking δ>2 and letting vn:=δc*un/tn, it follows that ∥vn∥2=δc* and there exists v∈X such that
{
v
n
⇀
v
weakly in
X
,
v
n
⇀
v
weakly in
H
s
(
ℝ
N
)
,
v
n
→
v
strongly in
L
loc
q
(
ℝ
N
)
,
2
≤
q
<
2
s
*
,
v
n
→
v
a.e. in
ℝ
N
.
We claim that v≠0. Indeed, since (vn) is bounded in X, we consider the quantity
lim inf
n
→
∞
(
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
v
n
|
2
𝑑
x
)
:=
α
.
Assume α=0. By Lemma 2.3 (passing to a subsequence, if necessary), vn→0 strongly in Lq(ℝN),q∈(2,2s*). Since (f1) and (f2) hold and ϵ>0 is arbitrary, we conclude that
∫
F
(
x
,
v
n
)
𝑑
x
≤
ϵ
(
∫
|
v
n
|
2
𝑑
x
+
∫
|
v
n
|
2
s
*
𝑑
x
)
+
C
ϵ
∫
|
v
n
|
p
𝑑
x
≲
ϵ
(
∥
v
n
∥
H
s
2
+
∥
v
n
∥
H
s
2
s
*
)
+
o
(
1
)
≲
ϵ
(
∥
v
n
∥
2
+
∥
v
n
∥
2
s
*
)
+
o
(
1
)
≤
ϵ
C
+
o
(
1
)
=
o
(
1
)
.
Thus,
Φ
(
v
n
)
=
1
2
∥
v
n
∥
2
-
∫
F
(
x
,
v
n
)
𝑑
x
=
δ
2
c
*
+
o
(
1
)
.
On the other hand, since un∈𝒩 and vn=δc*un/tn=:τnun, we have that
δ
2
c
*
+
o
(
1
)
=
Φ
(
v
n
)
≤
Φ
(
u
n
)
=
c
*
+
o
(
1
)
,
which is a contradiction since δ>2 by our choice. Therefore, we must have α>0, so that
(2.6)
∫
B
1
(
0
)
|
v
~
n
|
2
𝑑
x
≥
α
2
>
0
for n large, where v~n(x):=vn(x+zn) for some zn∈ℝN. That is,
(2.7)
v
~
n
(
x
)
:=
v
n
(
x
+
z
n
)
=
δ
c
*
t
n
u
~
n
(
x
)
,
u
~
n
(
x
)
:=
u
n
(
x
+
z
n
)
,
where we remember that we are still assuming (by contradiction) that tn=∥un∥→∞. Therefore, since ∥v~n∥=∥vn∥ is bounded, by passing to a subsequence if necessary, it follows that
{
v
~
n
⇀
v
~
weakly in
X
,
v
~
n
⇀
v
~
weakly in
H
s
(
ℝ
N
)
,
v
~
n
→
v
~
strongly in
L
loc
q
(
ℝ
N
)
,
2
≤
q
<
2
s
*
,
v
~
n
→
v
~
a.e. in
ℝ
N
,
where v~≠0 in view of (2.6).
Next, we divide (2.5) by tn2 to get
Φ
(
u
n
)
t
n
2
=
1
2
-
∫
F
(
x
,
u
n
)
t
n
2
𝑑
x
=
c
*
+
o
(
1
)
t
n
2
=
o
(
1
)
,
or, using translation invariance,
1
2
-
∫
F
(
x
,
u
~
n
)
t
n
2
𝑑
x
=
o
(
1
)
,
or still, by (2.7),
(2.8)
∫
F
(
x
,
u
~
n
)
u
~
n
2
v
~
n
2
𝑑
x
=
δ
c
*
2
+
o
(
1
)
.
However, in view of (f3) and v~≠0, we may conclude by Fatou’s lemma that
lim inf
n
→
∞
∫
F
(
x
,
u
~
n
)
u
~
n
2
v
~
n
2
𝑑
x
=
lim inf
n
→
∞
∫
F
(
x
,
t
n
v
~
n
/
δ
c
*
)
t
n
2
v
~
n
2
/
δ
c
*
v
~
n
2
𝑑
x
≥
∫
lim inf
n
→
∞
F
(
x
,
t
n
v
~
n
/
δ
c
*
)
t
n
2
v
~
n
2
/
δ
c
*
v
~
n
2
d
x
=
+
∞
,
which is a clear contradiction to (2.8). Thus, we conclude that (un) is bounded in X. ∎
Our next result yields a bounded Palais–Smale sequence for Φ at the level c*, which is a conclusion from Ekeland’s variational principle [13]. The proof is standard and we give it for completeness.
Lemma 2.7.
Assume (f1), (f2), and (f4). Given any minimizing sequence (un) for c*, there exists another minimizing sequence (v~n) such that ∥∇Φ(v~n)∥=o(1).
Proof.
Let (un) be a minimizing sequence for c*. Then (un) is bounded by Lemma 2.6 and, in view of Ekeland’s variational principle [13], there exists another minimizing sequence (vn) for c* such that
(2.9)
∥
v
n
-
u
n
∥
=
o
(
1
)
and
(2.10)
∥
[
∇
(
Φ
|
𝒩
)
]
(
v
n
)
∥
=
∥
P
T
v
n
𝒩
∇
Φ
(
v
n
)
∥
=
o
(
1
)
,
where PTv𝒩 denotes the orthogonal projection onto the tangent space to 𝒩 at v∈𝒩. That is,
P
T
v
𝒩
h
:=
h
-
〈
h
,
N
v
〉
N
v
,
where
h
∈
X
and
N
v
:=
∇
ψ
(
v
)
∥
∇
ψ
(
v
)
∥
is a unit normal to 𝒩 at v. Clearly, (2.9) implies that (vn) is also a bounded sequence. The rest of the proof is technical and consists in showing that
∥
∇
Φ
(
v
~
n
)
∥
=
o
(
1
)
for some other bounded minimizing sequence (v~n), as required. We start with the following claim.
Claim 1: ∥∇Φ(vn)∥ and ∥∇ψ(vn)∥ are bounded. Indeed, in view of (f1), (f2), and Sobolev’s inequality,
∥
∇
Φ
(
v
n
)
∥
X
*
=
sup
∥
v
n
∥
≤
1
〈
∇
Φ
(
v
n
)
,
v
n
〉
=
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
-
∫
f
(
x
,
v
n
)
v
n
𝑑
x
)
≤
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
|
∫
f
(
x
,
v
n
)
v
n
𝑑
x
|
)
≤
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
ϵ
(
∥
v
n
∥
L
2
2
+
∥
v
n
∥
L
2
s
*
2
s
*
)
+
C
ϵ
∥
v
n
∥
L
p
p
)
≲
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
ϵ
(
∥
v
n
∥
H
s
2
+
∥
v
n
∥
H
s
2
s
*
)
+
C
ϵ
∥
v
n
∥
H
s
p
)
.
Similarly, we have
∥
∇
ψ
(
v
n
)
∥
X
*
=
sup
∥
v
n
∥
≤
1
〈
∇
ψ
(
v
n
)
,
v
n
〉
=
sup
∥
v
n
∥
≤
1
(
2
∥
v
n
∥
2
-
∫
f
′
(
x
,
v
n
)
v
n
2
𝑑
x
-
∫
f
(
x
,
v
n
)
v
n
𝑑
x
)
≤
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
|
∫
f
′
(
x
,
v
n
)
v
n
2
𝑑
x
|
+
|
∫
f
(
x
,
v
n
)
v
n
𝑑
x
|
)
≤
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
ϵ
(
2
∥
v
n
∥
L
2
2
+
2
s
*
∥
v
n
∥
L
2
s
*
2
s
*
)
+
p
C
ϵ
∥
v
n
∥
L
p
p
)
≲
sup
∥
v
n
∥
≤
1
(
∥
v
n
∥
2
+
ϵ
(
2
∥
v
n
∥
H
s
2
+
2
s
*
∥
v
n
∥
H
s
2
s
*
)
+
p
C
ϵ
∥
v
n
∥
H
s
p
)
.
Therefore, claim 1 follows since (vn) is bounded in X, and so in Hs.
Next, taking the inner-product of PTvn𝒩∇Φ(vn) with ∇Φ(vn) and recalling that ∥∇Φ(vn)∥ is bounded, together with (2.10) we obtain
〈
∇
Φ
(
v
n
)
-
〈
∇
Φ
(
v
n
)
,
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
〉
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
,
∇
Φ
(
v
n
)
〉
=
o
(
1
)
.
That is to say,
(2.11)
∥
∇
Φ
(
v
n
)
∥
2
=
〈
∇
Φ
(
v
n
)
,
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
〉
2
+
o
(
1
)
.
Similarly, taking the inner product of PTvn𝒩∇Φ(vn) with vn and recalling that 〈∇Φ(vn),vn〉=ψ(vn)=0 (since vn∈𝒩), we obtain
-
〈
∇
Φ
(
v
n
)
,
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
〉
〈
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
,
v
n
〉
=
o
(
1
)
.
That is to say,
(2.12)
1
∥
∇
ψ
(
v
n
)
∥
⋅
|
〈
∇
Φ
(
v
n
)
,
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
〉
|
⋅
|
〈
∇
ψ
(
v
n
)
,
v
n
〉
|
=
A
n
B
n
C
n
=
o
(
1
)
.
We have the following claim.
Claim 2: Cn>0 is bounded from below away from zero. Indeed, since ∥vn∥2=∫ℝNf(x,vn)vn𝑑x and by (2.4), we may write
C
n
=
|
〈
∇
ψ
(
v
n
)
,
v
n
〉
|
=
|
2
∥
v
n
∥
2
-
∫
f
′
(
x
,
v
n
)
v
n
2
𝑑
x
-
∫
f
(
x
,
v
n
)
v
n
𝑑
x
|
=
∫
f
′
(
x
,
v
n
)
v
n
2
𝑑
x
-
∫
f
(
x
,
v
n
)
v
n
𝑑
x
.
Now, as in Lemma 2.6, we may consider the quantity
lim inf
n
→
∞
(
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
v
n
|
2
d
x
)
=
:
α
.
If α=0, by Lemma 2.3, then we have vn→0 in Lq(ℝN), q∈(2,2s*).
Note that
∫
F
(
x
,
v
n
)
𝑑
x
≤
ϵ
(
∥
v
n
∥
L
2
2
+
∥
v
n
∥
L
2
s
*
2
s
*
)
+
C
ϵ
∥
v
n
∥
L
p
p
=
o
(
1
)
and
∫
f
(
x
,
v
n
)
v
n
𝑑
x
≤
ϵ
(
∥
v
n
∥
L
2
2
+
∥
v
n
∥
L
2
s
*
2
s
*
)
+
C
ϵ
∥
v
n
∥
L
p
p
=
o
(
1
)
.
Thus,
c
*
+
o
(
1
)
=
Φ
(
v
n
)
=
1
2
∥
v
n
∥
2
-
∫
F
(
x
,
v
n
)
𝑑
x
=
1
2
∫
f
(
x
,
v
n
)
v
n
𝑑
x
-
∫
F
(
x
,
v
n
)
𝑑
x
=
o
(
1
)
.
This is a contradiction to c*>0. Thus, we must have α>0 and
(2.13)
∫
B
1
(
0
)
|
v
~
n
|
2
𝑑
x
≥
α
2
>
0
for n large, where v~n(x):=vn(x+zn) for some zn∈ℝN. By passing to a subsequence, if necessary, it follows that
(2.14)
{
v
~
n
⇀
v
~
weakly in
X
,
v
~
n
⇀
v
~
weakly in
H
s
(
ℝ
N
)
,
v
~
n
→
v
~
strongly in
L
loc
q
(
ℝ
N
)
,
2
≤
q
<
2
s
*
,
v
~
n
→
v
~
a.e. in
ℝ
N
,
where v~≠0 in view of (2.13). We have
C
n
=
∫
f
′
(
x
,
v
~
n
)
v
~
n
2
𝑑
x
-
∫
f
(
x
,
v
~
n
)
v
~
n
𝑑
x
by translation invariance. By assuming that Cn=o(1) and by Fatou’s lemma,
∫
lim inf
n
→
∞
(
f
′
(
x
,
v
~
n
)
-
f
(
x
,
v
~
n
)
v
~
n
)
v
~
n
2
d
x
≤
lim inf
n
→
∞
C
n
→
0
.
In view of (2.4), we conclude that v~=0, a contradiction. Thus, Cn is bounded from below away from zero.
By claim 1, we have ∥∇ψ(vn)∥≤C, and by claim 2 we have
∥
∇
ψ
(
v
n
)
∥
≥
1
∥
v
n
∥
|
〈
∇
ψ
(
v
n
)
,
v
n
〉
|
=
C
n
∥
v
n
∥
.
Thus, An is bounded away from zero.
Finally, we conclude from (2.12) that
B
n
=
|
〈
∇
Φ
(
v
n
)
,
∇
ψ
(
v
n
)
∥
∇
ψ
(
v
n
)
∥
〉
|
=
o
(
1
)
,
and from (2.11) and translation invariance that
∥
∇
Φ
(
v
~
n
)
∥
=
o
(
1
)
.
We have obtained a minimizing sequence (v~n)∈𝒩 for c* such that ∥∇Φ(v~n)∥=o(1), that is, a Palais–Smale sequence for Φ at the level c*:=limv∈𝒩Φ(v). ∎
Finally, we are in position to prove Theorem 1.1.
Proof.
Let (un) be a minimizing sequence for c*. In view of Lemma 2.6, (un) is bounded in X, and by Lemma 2.7, there exists another minimizing sequence (v~n)⊂𝒩 satisfying (2.14). In particular, we have
Φ
(
v
~
n
)
=
1
2
∫
(
f
(
x
,
v
~
n
)
v
~
n
-
2
F
(
x
,
v
~
n
)
)
𝑑
x
=
c
*
+
o
(
1
)
.
In view of (f4), [f(x,u)u-2F(x,u)]≥0, and Fatou’s lemma, we get
Φ
(
v
~
)
≤
lim inf
n
→
∞
Φ
(
v
~
n
)
=
c
*
.
That is,
(2.15)
1
2
∫
(
f
(
x
,
v
~
)
v
~
-
2
F
(
x
,
v
~
)
)
𝑑
x
≤
c
*
.
In addition, we have
Φ
′
(
v
~
n
)
⋅
θ
=
〈
v
~
n
,
θ
〉
-
∫
f
(
x
,
v
~
n
)
⋅
θ
𝑑
x
=
o
(
1
)
∥
θ
∥
for all θ∈Cc1(ℝN). Since v~n⇀v~ in X, we obtain 〈v~n,θ〉→〈v~,θ〉.
We may claim that
∫
f
(
x
,
v
~
n
)
⋅
θ
𝑑
x
→
∫
f
(
x
,
v
~
)
⋅
θ
𝑑
x
.
Indeed, letting Ω=suppθ, we have v~n→v~ strongly in Lp(Ω), 2≤p<2s*. By Lemma 2.2, there exist a subsequence (for simplicity, labeled (v~n)) and g∈Lp(Ω) such that
v
~
n
→
v
~
a.e. in
Ω
and
|
v
~
|
,
|
v
~
n
|
≤
g
a.e. in
Ω
.
By continuity,
f
(
x
,
v
~
n
)
⋅
θ
→
f
(
x
,
v
~
)
⋅
θ
a.e. in
Ω
,
and
|
f
(
x
,
v
~
n
)
⋅
θ
|
≤
sup
Ω
|
θ
|
⋅
|
f
(
x
,
v
~
n
)
|
≤
sup
Ω
|
θ
|
⋅
(
ϵ
(
|
v
~
n
|
+
|
v
~
n
|
2
s
*
-
1
)
+
C
ϵ
|
v
~
n
|
p
-
1
)
≤
sup
Ω
|
θ
|
⋅
(
ϵ
(
g
+
|
v
~
n
|
2
s
*
-
1
)
+
C
ϵ
g
p
-
1
)
∈
L
1
(
Ω
)
.
Then, by Lebesgue’s dominated convergence theorem,
∫
Ω
f
(
x
,
v
~
n
)
⋅
θ
𝑑
x
→
∫
Ω
f
(
x
,
v
~
)
⋅
θ
𝑑
x
,
which holds for all θ∈Cc1(ℝN). By density, it holds for all ω∈Hs(ℝN), so ω∈X. Therefore, we conclude that v~≠0 is a weak solution of (2.1), and it follows that v~∈𝒩. In particular, by the definition of c* and from (2.15), we also conclude that Φ(v~)=c*. This completes the proof. ∎
3 The Existence of Ground States in Potential Well Case
In this section, we shall consider the case when the potential function V(x) has a bounded potential well, and our goal is to prove Theorem 1.2.
One may show that the norm defined by
∥
u
∥
=
(
∫
ℝ
N
|
(
-
Δ
)
s
/
2
u
|
2
+
V
(
x
)
u
2
d
x
)
1
2
is equivalent to the standard norm on Hs. We may define the space X=Hs and we define 𝒩 etc. as before.
To prove Theorem 1.2, we do some preparations. We set V∞=lim|x|→∞V(x). There is an associated problem at infinity
(
-
Δ
)
s
u
+
V
∞
u
=
f
(
u
)
in
ℝ
N
.
Recall that the norm
∥
u
∥
∞
:=
(
∫
ℝ
N
(
|
(
-
Δ
)
s
/
2
u
|
2
+
V
∞
u
2
)
𝑑
x
)
1
2
is equivalent to ∥u∥Hs. Thus, it is also equivalent to ∥u∥. We define the energy functional
Φ
∞
=
1
2
∥
u
∥
∞
2
-
∫
ℝ
N
F
(
u
)
𝑑
x
and we define ψ∞(u)=〈Φ∞′(u),u〉. Moreover, c∞=inf𝒩∞Φ∞(u), where 𝒩∞={u∈X∖{0}:ψ∞(u)=0}. Since V∞ is a constant, by the argument of Theorem 1.1, c∞>0 is achieved at some u∞∈𝒩∞.
We now show that the level c∞ is larger than c*.
Lemma 3.1.
It holds 0<c*<c∞.
Proof.
Indeed, by the definition of c* and by (f4),
c
*
=
1
2
∥
u
∥
2
-
∫
F
(
u
)
=
1
2
∫
(
f
(
u
)
u
-
2
F
(
u
)
)
>
0
.
Recall that u∞ is the minimizer at c∞. Then ψ(u∞)<ψ∞(u∞)=0. By arguments similar to the ones in Lemma 2.5, there exists t^=t^(u∞)>0 such that ψ(t^u∞)=0, that is, t^u∞∈𝒩. Thus,
c
*
≤
Φ
(
t
^
u
∞
)
<
Φ
∞
(
t
^
u
∞
)
≤
Φ
∞
(
u
∞
)
=
c
∞
,
as desired. ∎
We note that, with minor changes, similar conclusions to Lemma 2.4, Lemma 2.5, and Lemma 2.7 still hold.
Lemma 3.2.
Let (un)⊂X such that ψ(un)→0 and ∫RNf(un)un→β>0. Then there exists tn>0 such that tnun∈N and tn→1.
Proof.
The proof is similar to the corresponding one in the paper of Li and Wang [21] and we omit the details. ∎
Lemma 3.3.
Assume (f1)–(f3). Then any minimizing sequence (un) for c* is bounded.
Proof.
Let (un)∈𝒩 be a minimizing sequence for c*. We argue by contradiction that ∥un∥→∞. Suppose vn:=un/∥un∥. Then ∥vn∥=1 and there exists v∈X such that
{
v
n
⇀
v
weakly in
X
,
v
n
⇀
v
weakly in
H
s
(
ℝ
N
)
,
v
n
→
v
strongly in
L
loc
q
(
ℝ
N
)
,
2
≤
q
<
2
s
*
,
v
n
→
v
a.e. in
ℝ
N
.
If vn→0 in Lq(ℝN),2≤q<2s*, then, by (f1), (f2), and Lebesgue’s dominated convergence theorem, for any s>0 we have ∫ℝNF(svn)𝑑x→0.
Then we have
c
*
+
o
(
1
)
=
Φ
(
u
n
)
≥
Φ
(
s
v
n
)
=
1
2
s
2
-
∫
ℝ
N
F
(
s
v
n
)
𝑑
x
=
1
2
s
2
+
o
(
1
)
.
We get a contradiction by choosing s>2c*.
Thus, vn↛0 in Lq(ℝN), 2≤q<2s*, and it follows from Lemma 2.3 that
(3.1)
∫
B
1
(
y
n
)
v
n
2
𝑑
x
≥
δ
for some δ>0, yn∈ℝN, and almost all n.
If the sequence (yn) is bounded, we may assume, after translating by (vn) if necessary, vn→v in Lloc2(ℝN), so (3.1) implies that v≠0. Thus, we get a contradiction by (f3) and Fatou’s lemma:
o
(
1
)
=
Φ
(
u
n
)
∥
u
n
∥
2
=
1
2
-
∫
F
(
u
n
)
∥
u
n
∥
2
=
1
2
-
∫
F
(
u
n
)
u
n
2
v
n
2
→
-
∞
.
If |yn|→∞ for a subsequence, similarly in view of (3.1), then v~n→v~≠0 in Lloc2(ℝN), where v~n=vn(⋅-yn) and u~n=un(⋅-yn). In view of (f3) and Fatou’s lemma,
o
(
1
)
=
Φ
(
u
n
)
∥
u
n
∥
2
=
1
2
-
∫
F
(
u
n
)
∥
u
n
∥
2
=
1
2
-
∫
F
(
u
n
)
u
n
2
v
n
2
=
1
2
-
∫
F
(
u
~
n
)
u
~
n
2
v
~
n
2
→
-
∞
.
This is a contradiction. Thus, (un) is bounded.
Assume that un⇀u in X and un→u in Llocp(ℝN). We claim that u≠0. Indeed, if u=0, in view of (V2), we have
∫
ℝ
N
(
V
(
x
)
-
V
∞
)
|
u
n
|
2
𝑑
x
→
0
.
Thus, we have
c
*
+
o
(
1
)
=
Φ
(
u
n
)
=
Φ
∞
(
u
n
)
+
o
(
1
)
and
ψ
∞
(
u
n
)
=
ψ
(
u
n
)
+
o
(
1
)
=
o
(
1
)
.
By Lemma 3.2, we have
c
*
+
o
(
1
)
=
Φ
∞
(
u
n
)
+
o
(
1
)
=
Φ
∞
(
t
n
u
n
)
+
o
(
1
)
≥
c
∞
+
o
(
1
)
,
a contradiction to Lemma 3.1. Thus we have u≠0, and the proof is complete. ∎
We are now in position to prove Theorem 1.2.
Proof.
Let (un) be a minimizing sequence for c*. Then, by Lemma 3.3, (un) is bounded in X and un⇀u≠0. By Lemma 2.6, there is another minimizing sequence (v~n) satisfying (2.14). By definition, we have
Φ
(
v
~
n
)
=
1
2
∫
[
f
(
v
~
n
)
v
~
n
-
2
F
(
v
~
n
)
]
𝑑
x
=
c
*
+
o
(
1
)
.
Hence, by [f(u)u-2F(u)]≥0 and Fatou’s lemma, we have
1
2
∫
[
f
(
v
~
)
v
~
-
2
F
(
v
~
)
]
𝑑
x
≤
c
*
.
In addition,
Φ
′
(
v
~
n
)
⋅
θ
=
〈
v
~
n
,
θ
〉
-
∫
f
(
v
~
n
)
θ
=
o
(
1
)
∥
θ
∥
for all θ∈Cc1(ℝN). Remembering (2.14) by weak convergence, we have 〈v~n,θ〉→〈v~,θ〉. In view of (f1), (f2), and Lemma 2.2, by Lebesgue’s dominated convergence theorem, we have
∫
supp
θ
f
(
v
~
n
)
θ
→
∫
supp
θ
f
(
v
~
)
θ
.
Hence,
Φ
′
(
v
~
)
⋅
θ
=
0
for all θ∈Cc1(ℝN). By density,
Φ
′
(
v
~
)
⋅
ω
=
0
for all ω∈X. Then we conclude that v~≠0 is a critical point of Φ and hence a weak solution of (1.3). In particular, Φ(v~)=c*. This completes the proof. ∎
and
Zhenxiong Li
